Number Theory and the Real Number System CHAPTER 5 Number Theory and the Real Number System

5.3 The Rational Numbers

Objectives Define the rational numbers. Reduce rational numbers. Convert between mixed numbers and improper fractions. Express rational numbers as decimals. Express decimals in the form a / b. Multiply and divide rational numbers. Add and subtract rational numbers. Use the order of operations agreement with rational numbers. Apply the density property of rational numbers. Solve problems involving rational numbers.

Defining the Rational Numbers The set of rational numbers is the set of all numbers which can be expressed in the form , where a and b are integers and b is not equal to 0. The integer a is called the numerator. The integer b is called the denominator. The following are examples of rational numbers: ¼, ½, ¾, 5, 0

Reducing a Rational Number If is a rational number and c is any number other than 0, The rational numbers and are called equivalent fractions. To reduce a rational number to its lowest terms, divide both the numerator and denominator by their greatest common divisor.

Example: Reducing a Rational Number Reduce to lowest terms. Solution: Begin by finding the greatest common divisor of 130 and 455. Thus, 130 = 2 · 5 · 13, and 455 = 5 · 7 · 13. The greatest common divisor is 5 · 13 or 65.

Example: Reducing a Rational Number (continued) Divide the numerator and the denominator of the given rational number by 5 · 13 or 65. There are no common divisors of 2 and 7 other than 1. Thus, the rational number is in its lowest terms.

Mixed Numbers and Improper Fractions A mixed number consists of the sum of an integer and a rational number, expressed without the use of an addition sign. Example: An improper fraction is a rational number whose numerator is greater than its denominator. 19 is larger than 5

Converting a Positive Mixed Number to an Improper Fraction Multiply the denominator of the rational number by the integer and add the numerator to this product. Place the sum in step 1 over the denominator of the mixed number.

Example: Converting a Positive Mixed Number to an Improper Fraction Convert to an improper fraction. Solution:

Converting a Positive Improper Fraction to a Mixed Number Divide the denominator into the numerator. Record the quotient and the remainder. Write the mixed number using the following form:

Example: Converting from an Improper Fraction to a Mixed Number Convert to a mixed number. Solution: Step 1 Divide the denominator into the numerator. Step 2 Write the mixed number using Thus,

Rational Numbers and Decimals Any rational number can be expressed as a decimal by dividing the denominator into the numerator.

Example: Expressing Rational Numbers as Decimals Express each rational number as a decimal. a. b. Solution: In each case, divide the denominator into the numerator.

Example: Expressing Rational Numbers as Decimals (continued) Notice the digits 63 repeat over and over indefinitely. This is called a repeating decimal. Notice the decimal stops. This is called a terminating decimal.

Expressing Decimals as a Quotient of Two Integers Terminating decimals can be expressed with denominators of 10, 100, 1000, 10,000, and so on. Using the chart, the digits to the right of the decimal point are the numerator of the rational number.

Example: Expressing Decimals as a Quotient of Two Integers Express each terminating decimal as a quotient of integers: a. 0.7 b. 0.49 c. 0.048 Solution: 0.7 = because the 7 is in the tenths position.

Example: Expressing Terminating Decimals in a/b form (continued) b. 0.49 = because the digit on the right, 9, is in the hundredths position. c. 0.048 = because the digit on the right, 8, is in the thousandths position. Reducing to lowest terms,

Example: Expressing a Repeating Decimal in a/b Form Express as a quotient of integers. Solution: Step 1 Let n equal the repeating decimal such that n = , or 0.6666… Step 2 If there is one repeating digit, multiply both sides of the equation in step 1 by 10. n = 0.66666… 10n = 10(0.66666…) 10n = 6.66666… Multiplying by 10 moves the decimal point one place to the right.

Example: Expressing a Repeating Decimal in a/b Form (continued) Step 3 Subtract the equation in step 1 from the equation in step 2. Step 4 Divide both sides of the equation in step 3 by the number in front of n and solve for n. We solve 9n = 6 for n: Thus, .

Multiplying Rational Numbers The product of two rational numbers is the product of their numerators divided by the product of their denominators. If and are rational numbers, then

Example: Multiplying Rational Numbers Multiply. If possible, reduce the product to its lowest terms: Simplify to lowest terms. Multiply across.

Dividing Rational Numbers The quotient of two rational numbers is a product of the first number and the reciprocal of the second number. If and are rational numbers, and c/d is not 0, then

Example: Dividing Rational Numbers Divide. If possible, reduce the quotient to its lowest terms: Change to multiplication by using the reciprocal. Multiply across.

Adding and Subtracting Rational Numbers with Identical Denominators The sum or difference of two rational numbers with identical denominators is the sum or difference of their numerators over the common denominator. If and are rational numbers, then and

Perform the indicated operations: a. b. c. Solution: Example: Adding & Subtracting Rational Numbers with Identical Denominators Perform the indicated operations: a. b. c. Solution:

Adding and Subtracting Rational Numbers Unlike Denominators If the rational numbers to be added or subtracted have different denominators, we use the least common multiple of their denominators to rewrite the rational numbers. The least common multiple of their denominators is called the least common denominator or LCD.

Example: Adding Rational Numbers Unlike Denominators Find the sum of Solution: Find the least common multiple of 4 and 6 so that the denominators will be identical. LCM of 4 and 6 is 12. Hence, 12 is the LCD. We multiply the first rational number by 3/3 and the second one by 2/2 to obtain 12 in the denominator for each number. Notice, we have 12 in the denominator for each number. Add numerators and put this sum over the least common denominator.

Density of Rational Numbers If r and t represent rational numbers, with r < t, then there is a rational number s such that s is between r and t: r < s < t.

Example: Illustrating the Density Property Find a rational number halfway between ½ and ¾. Solution: First add ½ and ¾. Next, divide this sum by 2. The number is halfway between ½ and ¾. Thus,

Problem Solving with Rational Numbers A common application of rational numbers involves preparing food for a different number of servings than what the recipe gives. The amount of each ingredient can be found as follows:

Example: Adjusting the Size of a Recipe A chocolate-chip recipe for five dozen cookies requires ¾ cup of sugar. If you want to make eight dozen cookies, how much sugar is needed? Solution:

Example (continued) The amount of sugar, in cups, needed is determined by multiplying the rational numbers: Thus, cups of sugar is needed.